846
Ind. Eng. Chem. Res. 2007, 46, 846-853
Null Space Method for Selecting Optimal Measurement Combinations as Controlled Variables Vidar Alstad† and Sigurd Skogestad* Department of Chemical Engineering, Norwegian UniVersity of Science and Technology, Trondheim, Norway
The issue in this paper is to select controlled variables c as combinations of the measurements y. The objective is to obtain self-optimizing control, which is when we can achieve near-optimal steady-state operation with constant setpoints for the controlled variables, without the need to reoptimize when new disturbances perturb the plant. The null space method yields locally optimal controlled variables c ) Hy that are linear combinations of measurements y. The requirement is that we at least have as many measurements as there are unconstrained degrees of freedom, including disturbances, and that the implementation error is neglected. The method is surprisingly simple. From a steady-state model of the plant, the first step is to obtain the optimal sensitivity matrix F, with respect to the disturbances. The optimal matrix H satisfies HF ) 0; therefore, the next step is to obtain H in the left null space of F. As an illustration, the method is used to obtain temperature combinations for control of a Petlyuk distillation column. 1. Introduction For cases with unconstrained control degrees of freedom (inputs u), an important issue is to decide what to control. Although not widely acknowledged by control theorists, controlling the right variables is a key element in overcoming uncertainty in operation.1,2 This applies also when using advanced control (e.g., model predictive control (MPC)) or realtime optimization (RTO). This paper focuses on the interaction between the local optimization layer and the feedback control layer (see Figure 1), and, more specifically, on the selection of the controlled variables c that link these layers. Two subproblems are important here: (1) Selection of the controlled Variables c. This is a structural decision that is made before the control strategy is implemented. (2) Selection of setpoints cs. This is a parametric decision that can be done both online and offline. Here, we focus on the first structural problem of finding the controlled variables and we will assume constant nominal optimal setpoints. As Figure 1 shows, there are two sources of uncertainty that will make a constant setpoint policy nonoptimal: (i) disturbances (d) (these are the external unmeasured disturbances, including parameter variations), and (ii) implementation error (n) (this includes the sum of the effect of the measurement error for y and the control error). Single measurements or functions or combinations of the measurements may be used as controlled variables (c). The objective is to obtain self-optimizing control, which is when we can achieve near-optimal steady-state operation with constant setpoints for the controlled variables, without the need to reoptimize when new disturbances perturb the plant. The use of single measurements is simple and is the preferred choice if the loss is sufficiently small. However, for some applications, there may not exist any self-optimizing single measurements, and one may consider measurement combinations. In this paper, * To whom correspondence should be addressed. Tel.: +47 73 59 41 54. Fax: +47 73 59 40 80. E-mail:
[email protected]. † Current affiliation: Norsk Hydro ASA, Oil & Energy Research Center, N-3908 Porsgrunn, Norway.
Figure 1. Block diagram of a feedback control structure including an optimizer layer.
we consider linear combinations, that is, c ) Hy, where H is a constant matrix. Ideas related to self-optimizing control have been presented repeatedly in the process control history, but the first quantitative treatment was that of Morari et al.1 Skogestad2 defined the problem more carefully, linked it to previous work, and was the first to include the implementation error as well. He mainly considered the case where single measurements are used as controlled variables; that is, H is a selection matrix where each row has a single digit (one, 1) and the rest are zeros (0). The loss with a constant setpoint policy for expected disturbances and implementation errors was evaluated using a “brute-force’’ approach. An important advantage of a brute-force evaluation is that one can also identify controlled variables that may yield infeasability for certain disturbances or implementation errors. This was also considered in more detail by Larsson et al.3 for the Tennessee-Eastman challenge problem and Govatsmark and Skogestad,4 who suggested adjustment of the setpoints to achieve feasibility. However, the computational load of the “brute-force’’ method can be very large, so local methods based on linearizing the behavior around the steady-state are attractive. Skogestad2 introduced the approximate maximum gain rule as a simple method for selecting controlled variables. In the
10.1021/ie060285+ CCC: $37.00 © 2007 American Chemical Society Published on Web 01/03/2007
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 847
multivariable case, the gain is the minimum singular value of the scaled steady-state transfer matrix from u to c. A similar method was presented by Mahajanam et al.5 Halvorsen et al.6 considered the maximum gain method in more detail and also proposed an exact local method that may be used to obtain the optimal measurement combination H. However, this method is less attractive computationally and, in addition, is somewhat difficult to use. Hori et al.7 illustrated the ideas introduced in this paper on indirect control, which can be formulated as a subproblem of the null space method that has been presented in this paper. Related work has been done by Srinivasan8-10 on measurement-based optimization to enforce the necessary condition of optimality under uncertainty. The ideas are illustrated on batch processes. Francois et al.11 extended these ideas and focus on steady-state optimal systems, where a clear distinction is made between enforcing active constraints and requiring the sensitivity of the objective to be zero. Guay and Zhang12 have presented related ideas on measurement-based dynamic optimization. In this paper, the objective is to derive a simple method for selecting the optimal measurement combination matrix H for the special case with no implementation error. In fact, the method is so simple that the second author (Skogestad) thought it had to be wrong when it was proposed by the first author (Alstad). We have attempted to keep the mathematics as simple as possible. A more-detailed comparison with previous results and extensions are presented in a forthcoming publication (also see ref 13). 2. Problem Formulation We assume that the operational goal is to use the degrees of freedoms (u) to minimize the cost (J), while satisfying equality and inequality constraints. The (original) constrained steadystate optimization problem can, for a given disturbance d, be formulated as
min J(x, u, d) x,u
(1)
subject to
g(x, u, d) e 0 (2)
where x ∈ Rnx, u ∈ Rnu, and d ∈ Rnd are the states, inputs, and disturbances, respectively; f is the set of equality constraints corresponding to the model equation; g is the set of inequality constraints that limits the operation (e.g., physical limits on temperature measurements or flow constraints); and y represents the measurements. When solving this optimization problem, we generally find that some constraints are active, corresponding to gi(x, u, d) ) 0. To achieve steady-state optimal operation, we assume here that we control all the active constraints14,15 and that the set of active constraints does not change (see assumption A3 presented later in this work). Controlling the active constraints consumes a corresponding number of the degrees of freedom (inputs), and we consider, in the following, the remaining unconstrained reduced-space optimization problem where the scalar cost function J is to be minimized with respect to the nu remaining degrees of freedom (inputs) u:
min J(u, d) u
1 L ) J(c, d) - J(copt(d),d) ≈ (c - copt)TJcc(c - copt) 2
(3)
(4)
where the second-order approximation holds for small deviations from the nominal optimum. The selected controlled variables c are assumed to be independent and the Hessian matrix Jcc is assumed to be nonsingular.6 With a constant setpoint policy, we have c ) cs + n, where n is the implementation error. In this paper, we assume n ) 0 (see assumption A4 presented later in this work) and assume that the setpoint is nominally optimal, cs ) copt(d*), where d* is the nominal value of the disturbance. Then, c ) copt and the loss for small deviations from the nominal optimum is
1 L ≈ (copt(d*) - copt(d))TJcc(copt(d*) - copt(d)) 2
f(x, u, d) ) 0 y ) fy(x, u, d)
Here, the equality constraints, including the model equations and active constraints, are implicitly included in J, so J is generally not a simple function of u and d. Remark. Although not strictly correct, for simplicity, we use the same symbol for the cost J and degrees of freedom u in expressions 1 and 3. Also note that it does not matter from a steady-state point of view which of the original degrees of freedom are used to satisfy the active constraints, and which are then left in expression 3 as the unconstrained degrees of freedom u, as long as the remaining optimization problem remains well-posed. We assume that online information about the system behavior is available through measurements y. The issue in this paper is finding a set of nu controlled variables c ) h(y) associated with the “unconstrained’’ degrees of freedom u. In the measurement vector y, we generally include also the input vector u, including the inputs u′ that have been selected to the control active constraints. However, the measurements of the active constraints are not included in y, because they are constant and, thus, provide no information about the operation. To quantify the difference between alternative control policies, we consider the loss. For a given disturbance d, the loss is defined as the difference between the actual cost and the optimal cost.6
(5)
This gives the following insight (which is not very surprising and could have been stated directly): With independent controlled Variables c and no implementation error, a constant setpoint policy is optimal if copt(d) is independent of d, i.e. copt(d) - copt(d*) ) 0. 3. Null Space Method We consider the unconstrained optimization problem as given by eq 3; that is, we assume “active constraint control”, where all optimally constrained variables are assumed to be kept constant at their optimal values. The goal is to find a linear measurement combination c ) Hy to be kept at constant setpoints cs. Here, H is a constant nu × ny matrix and y is a subset of the available measurements. We make the following assumptions: A1. Steady state: We consider only steady-state operation. The justification for this is that the economics of operation is primarily determined by the steady state. Of course, this assumes that we have a control system in place that can quickly bring the plant to its new steady state. A2. Disturbances: Only disturbances that affect the steadystate operation are included.
848
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
A3. ActiVe constraint control: We assume that the same active constraints remain active for all values of the disturbances and that we control these constraints. A4. No implementation error: The implementation error is the sum of the control error and the effect of the measurement error. The assumption of no steady-state control error is satisfied if we use a controller with integral action. It is a more serious assumption to neglect the measurement error, so the method implicitly assumes that the measurements have been carefully selected. We then have the following result. Theorem 1: Null space method. Assume that we haVe nu independent unconstrained free Variables u, nd independent disturbances d, ny independent measurements y, and we want to obtain nc ) nu independent controlled Variables c that are linear combinations of the measurements
c ) Hy
(6)
Let
F)
∂yopt ∂dT
be the optimal sensitiVity matrix eValuated with constant actiVe constraints. If ny g nu + nd, it is possible to select the matrix H in the left null space of F, H ∈ N(FT), such that we get
HF ) 0 With this choice for H, fixing c (at its nominal optimal Value) is first-order optimal for disturbances d; that is, the loss is zero as long as the sensitiVity matrix F does not change. Proof. We first prove that selecting H such that HF ) 0 gives zero disturbance loss. For small disturbances, the optimal change in the measurements to a change in the disturbances can be written as
where
yopt(d) - yopt(d) ) F(d - d*)
(7)
∂yopt ∂yopt 1 1 ‚‚‚ ∂dnd ∂d1 ‚‚‚ F) ∂ynopt ∂ynopt y y ‚‚‚ ∂dnd ∂d1
(8)
[ ]
is the optimal sensitivity matrix evaluated at the nominal point *. In eq 7, we have only included the first-order term in the Taylor expansion, so eq 7 is valid for small disturbances such that second- and higher-order terms in (d - d*) can be neglected, or, equivalently, as long as F does not change. From eq 6, the corresponding optimal change in the controlled variables is copt(d) - copt(d*) ) H(yopt(d) - yopt(d*)) and by inserting eq 7 we get
copt(d) - copt(d) ) HF(d - d*)
(9)
From the insight stated at the end of the previous section, the constant setpoint policy is optimal if
copt(d) - copt(d*) ) 0
(10)
which gives the requirement
HF(d - d*) ) 0
(11)
This must be satisfied for any (d - d*), so we must require that
HF ) 0
(12)
Next, we must prove the conditions under which this is possible. To satisfy HF(d - d*) ) 0, we must select H such that H ∈ N(FT). The rank of the nc × ny matrix H is nu (because ny g nc, nc ) nu, and the controlled variables are independent). The rank of the ny × nd matrix F is nd (because ny g nd and the disturbances are assumed to be independent). The fundamental theorem of linear algebra says that the left null space of F (N(FT)) has a rank of ny - r, where r ) nd is the rank of F. To be able to find an H of rank nu in the left null space of F, we must then require ny - nd g nu or, equivalently, ny g nu + nd. 0 The proposed null space method is optimal only locally. It is globally optimal in cases where the sensitivity matrix F is not dependent on the operating point (disturbances), for example, for a system with a quadratic cost objective and linear model equations (see eq 13, presented later in this work). Nevertheless, based on several case studies, this does not seem to be an important limitation in most practical cases. Obtaining F. The optimal sensitivity matrix F may be computed from the Hessian matrices (Juu and Jud) and steadystate gain matrices (Gy and Gyd), using6 y F ) -(GyJ-1 uu Jud - Gd)
(13)
However, in practice, it is usually easier to obtain F directly, by optimizing for the selected disturbances using a nonlinear steady-state model of the plant. Note that we do not necessarily need an explicit representation of the model equations, because we can find F numerically. For example, we may use one of the commercial steady-state process simulators, such as Aspen Plus or Hysys. In theory, one may even obtain F from experiments on a real operating plant; however, it seems unlikely that this methodology will be sufficiently accurate. Numerically, the ny × nd matrix F may be obtained by perturbing the disturbances d and re-solving the optimization problem in eq 3 with the active constraints being constant: (1) Under nominal conditions (d ) d*), use the steady-state model to obtain the nominal optimum yopt(d*) and identify the active constraints (finding the nominal optimum may be difficult, because the optimization problem is generally nonconvex). (2) For each of the nd disturbances, make a small perturbation (dk ) d* K + ∆dk) and resolve the optimization with the constant active constraints to obtain yopt(d) (this is generally simple, because it is only a small perturbation to the existing nominal solution). (3) Compute ∆yopt ) yopt(d) - yopt(d*) and obtain F numerically using eq 8. A minimum of nd + 1 optimization runs are required; however, more runs may be needed to obtain an accurate estimate for F. Ganesh and Biegler17 have provided an efficient and rigorous strategy for finding F, based on a reduced Hessian method. In addition, some process simulators have built-in optimizers from which the optimal sensitivity F may be available. The next step is to obtain H. Numerically, H may be obtained from a singular value decomposition of FT. We have HF ) 0 or, equivalently, FTHT ) 0. Thus, selecting HT as the input
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 849
singular vectors of FT, corresponding to zero singular values in FT, gives an orthogonal basis. Example 1. Consider a simple example with one unconstrained degree of freedom u , nu ) 1, and one disturbance d, nd ) 1. The cost function to be minimized during operation (for Varying d) is
J(u, d) ) (u - d)
2
Nominally, d* ) 0. We haVe aVailable two measurements:
y1 ) 0.9u + 0.1d and
y2 ) 0.5u - d Because ny ) 2 ) nu + nd and the two measurements are independent, it is possible to find a linear measurement combination
[]
y c ) Hy ) [h1 h2 ] y1 ) h1y1 + h2y2 2 for which a constant setpoint giVes zero disturbance loss, at least locally. We first must obtain the optimal sensitiVity matrix F. Optimality is ensured when (∂J/∂u) ) 2(u - d) ) 0, which giVes uopt ) d and Jopt ) 0 ∀ d. The corresponding optimal outputs are
yopt 1 )d and
yopt 2 ) -0.5d and (8) giVes that FT ) [1 -0.5]. AlternatiVely, use (13) with GYT ) [0.9 0.5], GYT d ) [0.1 -1], Juu ) 2, and Jud ) -2. From the null space method, the optimal matrix H must satisfy HF ) 0, or
h1f1 + h2f2 ) 0 h1 + h2(-0.5) ) 0 w h1 ) 0.5h2 The solution is non-unique. For example, selecting h2 ) 1 giVes
c ) 0.5y1 + y2 Keeping the controlled Variable c at its nominally optimal setpoint, cs ) copt(d*) ) 0, giVes zero disturbance loss, as is easily Verified. Generally, the loss will be zero only locally, i.e., for small changes in d, but, for this example, the cost function is quadratic with linear model equations, and the loss will be zero for any magnitude of the disturbance d. 4. Discussion 4.1. Measurement Selection. One weakness of the null space method is that it does not consider the measurement error, or, more generally, the implementation error. If we have extra measurements, that is, ny > nu + nd, then we have extra degrees of freedom in selecting H that should be used to reduce the sensitivity to measurement error. A simple approach is to select a subset of the “best’’ measurements such that we get ny ) nu + nd, but which should these measurements be? This is outside the scope of this paper, and it will be treated in more detail a forthcoming publication on the extended null space method (also
see Alstad13), but let us provide some results. Let the linear model be
∆y ) Gy∆u + Gyd∆d ) G ˜y
[ ] ∆u ∆d
(14)
where y has been scaled, with respect to the expected measurement error, and u and d have been scaled such that they have similar effects on the cost. It can then be shown that a reasonable approach is to maximize the minimum singular value of the matrix G ˜ y ) [GyGyd] from the combined inputs and disturbances to the selected measurements. To understand why this is reasonable, we may imagine using the measurements to backcalculate the inputs and disturbances. For the case with ny ) nu + nd , G ˜ y is invertible and we get
[ ]
∆u ) [G ˜ y]-1∆y ∆d
(15)
To avoid sensitivity to measurement errors in y, we want the norm of [G ˜ y]-1 to be small, which is equivalent to wanting a large minimum singular value, σ(G ˜ y). From eq 15, we also see why it is reasonable to require ny g nu + nd in the null space method, because this is the requirement for being able to uniquely determine all independent variables (inputs and disturbances) from the measurements. 4.2. Freedom in Selecting H. Even for the case ny ) nu + nd, there are an infinite number of matrices H that satisfy HF ) 0. This stems from the freedom of selecting basis vectors for the null space. Let H0 be one such matrix, i.e., H0F ) 0. For example, H0 may consist of the one set of basis vectors that span the null space of FT. Then, H ) CH0 also satisfies HF ) 0, provided the nc × nc matrix C is nonsingular. Actually, the degrees of freedom in selecting C (and H) are the same as the degrees of freedom that are used in steadystate decoupling (or similar) in control. The linear model for the selected controlled variables can be written as
∆c ) H∆y ) HGy∆u + HGyd∆d ) G∆u + Gd∆d
(16)
and the degrees of freedom in the matrix C may be used to affect G ) HGy and Gd ) HGyd. For example, it is possible to select H such that G ) I, and we have a decoupled steadystate response from u to c. 4.3. Disturbances. The minimum number of measurements required in the null space method (nu + nd) may be large if we have many disturbances (a large nd value). Therefore, in practical applications, it is desirable to reduce the number of disturbances. Unfortunately, there does not seem to be any simple rigorous procedure for eliminating unimportant disturbances, although some approaches are discussed in Chapter 5 of the work by Alstad.13 It is obvious that we may eliminate disturbances di that satisfy both of the following conditions: (1) no steadystate effect on the measurements (y is independent of di, i.e., Gdiy ) ), and (2) no steady-state effect on the optimal operation (uopt is independent of di). It could be argued that we may eliminate all “unobservable’’ disturbances that satisfy condition 1, because we have no way of detecting them and, thus, correcting for them. However, such disturbances may affect the optimal operation and result in large losses; therefore, an analysis based on neglecting them may be highly misleading. To achieve acceptable operation in such cases, we must obtain additional measurements, for example, of the disturbance itself. One example would be a price change, as is discussed in more detail below. Also, we cannot eliminate
850
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
all disturbances that have no effect on optimal operation and thus satisfy condition 2. This is because the disturbance may effect a measurement, and controlling this measurement will then result in a loss. In practice, with too few measurements, one may eliminate some disturbances and obtain the controlled variables c ) Hy using the null space method, but, afterward, one should analyze the loss with all disturbances included. Alternatively, one may be able to obtain the optimal combination numerically using the exact local method of Halvorsen et al.6 or the extended null space method that will be presented in a forthcoming publication. 4.4. Physical Interpretation. The proposed null space method yields controlled variables that are linear combinations of the available measurements. A disadvantage is that the physical interpretation of what we control is usually lost. This is by no means a fundamental limitation, because, in principle, we can control any signal from the process, as long as it is independent. Thus, if all measurements are regarded as signals, the concept of controlling a combination of signals may be easier to grasp. If possible, one can choose to combine measurements of one type, for instance, only temperatures (e.g., in a distillation column) or only mass flows. In any case, we can scale variables such that the resulting measurements are dimensionless, which is common in practice. 4.4.1. Change in Active Constraints. It has been assumed that the set of active constraints does not change. If they may change, then one must, for each set of active constraints, obtain new controlled variables c ) Hy satisfying HF ) 0, where F is evaluated with the given set of active constraints. This involves offline calculations. In addition, one must have an online strategy for identifying a change in active constraints change and some logic for switching to a new set of controlled variables. Thus, for cases where the active constraints shift frequently with the disturbances, other online methods may be better suited, for example, real-time optimization (RTO) combined with model predictive control (MPC). Alternatively, we could use the ideas of Arkun and Stephanopoulos15 in regard to how to handle varying active constraints. 4.5. Nonobservable Disturbances and Price Changes. Selfoptimizing control is based on using feedback to detect disturbances and optimally adjust the inputs to achieve nearoptimal operation. Thus, one must require that the disturbances are observable (visible) in the measurements y. One example of a “disturbance’’ that is not visible in the measurements is prices. However, prices (pi) do enter in the objective function, because typically J ) ∑ipixi, and price changes will change the optimal point of operation. To handle price changes (or, more generally, disturbances that are not observable in the measurements y), one must assume that the price (disturbances) is known (measured). Price changes can then be handled in two ways: (1) Adjust the setpoints in a feedforward manner. Then, for a price change ∆p, we have that
cs ) cs(p*) + HFp∆p
(17)
where Fp ) (∂yopt/∂pT) is the optimal sensitivity from the prices to the measurements. (2) Include the prices as extra measurements in y and use the regular procedure of selecting self-optimizing control variables, as previously described. The first approach is probably the simplest and most transparent.18
Figure 2. Schematic of the Petlyuk distillation column implemented in a single shell (“divided wall column”).
4.6. Limitations. As already mentioned, the proposed null space method is optimal only locally. It is globally optimal in cases where the sensitivity matrix F is not dependent on the operating point. Nevertheless, based on several case studies, this does not seem to be an important limitation in most practical cases. Other possibly more-serious limitations for practical use are that (1) implementation errors are not explicitly handled (except through the selection of which measurements to use), (2) the active constraints are assumed not to change (as discussed previously, in more detail), (3) the optimal setpoint for c must be obtained (not really a limitation, but rather a fact), and (4) the derivation of the optimal measurement combination c ) Hy is based on steady-state models, and one must later check that the resulting structure has acceptable dynamic controllability (this may usually be achieved using a cascade control structure where local controllers handle the dynamic response and c is controlled at steady state using the setpoints for the local controllers as manipulated variables). 5. Petlyuk Distillation Case Study 5.1. Introduction. The Petlyuk distillation column is an appealing alternative for the separation of ternary mixtures. Compared with the traditional configuration of two columns in series, typical savings on the order of 30% are reported in both energy and capital costs.19 However, the savings in energy may be difficult to achieve in practice, and the goal here is to suggest simple control policies. We are looking for a “self-optimizing” control structure that, despite external disturbances and measurements errors, gives near-optimal operation with constant setpoints. The Petlyuk column has six sections and may be implemented as a “divided wall’’ column, as illustrated in Figure 2. The boilup and reflux streams are split at the dividing wall with split fractions RV ) V2/V6 and RL ) L1/L3, respectively. With a given feed and pressure, the Petlyuk column has five steady-state degrees of freedom. For example, these may be selected as
uTall ) [L V S RL RV ]
(18)
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007 851 Table 1. Data for Petlyuk Case Study parameter
value
Column Data number of stages in each section NT ) 8 relative volatilities RT ) [9 3 1] boiling point A, B, and C [K] TTB ) [299.3 342.15 399.3] Feed flow composition liquid fraction
Product Compositions 0 xA,D * ) 0.97 0 xB,S* ) 0.97 0 xC,B * ) 0.97
distillate side-stream bottom feed flow feed composition zA zB liquid fraction product specification 0 xA,D 0 xC,B 0 xB,S temperatures flows RL, RV
F* ) 1 z*T ) [1/3 1/3 1/3] q* ) 0.477
Disturbances (d) F ) F* ( 0.1 zA ) z/A ( 0.1 zB ) z/B ( 0.1 q ) q* ( 0.1 0 0 * xA,D ) xA,D ( 0.01 0 0 xC,B ) xC,B / ( 0.01 0 0 xB,S ) xB,S * ( 0.01
Measurement/Implementation Errors (n) 0.5 K (absolute) 2.5% (relative) 0.025 absolute
corresponding to the reflux, boilup, side-stream flow, liquid split, and vapor split, respectively. Assume that the feed consists of three key components A, B, and C, with mole fractions zT ) [zA zB zC], a molar flow rate F, and a liquid fraction q. The light component A dominates in the distillate stream (D) and component B dominates in the side-stream (S), whereas the heavy component C dominates in the bottom stream (B). We consider a case study with a relative volatility of 3 between the key components and eight theoretical stages in each of the six sections. Key data are given in Table 1, and further details are found in Chapter 8 in the work by Alstad.13 We assume that the operational objective is to use the five degrees of freedom to minimize the energy usage, J ) V, while maintaining the following three product specifications (“active constraints’’): (1) distillate purity (xA,D), (2) bottom purity (xC,B), and (3) side-stream purity (xB,S), where xi,j is the mole fraction of component i in stream j. Minimizing the energy (V), with respect to the remaining two degrees of freedom, gives an unconstrained nominal optimum, with T R*L R*V ] ) uopt all (d*) ) [L* V* S* [0.7618 0.5811 0.3227 0.3792 0.5123 ]
The minimum boilup (Vmin) with an infinite number of stages is V∞min ) 0.5438; therefore, the nominal optimal boilup of 0.5811 is ∼6% higher than the theoretical minimum. Because the objective is to minimize the boilup, which also is an input, one may mistakenly believe that one can use an open-loop approach, where the optimal value for the boilup is calculated and implemented in the column, V ) Vopt. However, Halvorsen and Skogestad20 noted that such an approach is impossible (or at least very difficult): (1) The operation is infeasible for V < Vopt, so we must ensure that V g Vopt.
Figure 3. Physical location of the best subset of measurements for alternative 1.
Table 2. Alternative Control Structures Alternative
c1
c2
c3
c4
c5
1
xA,D
xB,S
xC,B
c1tdf
c2tdf
2 3 4
xA,D xA,D xA,D
xB,S xB,S xB,S
xC,B xC,B xC,B
RV RV RV
codf DTS RL
null space method, use RV and RL null space method, fix RV fix DTS and RV constant splits RV and RL
(2) The optimal value of V varies, with respect to the disturbances, and may be difficult to find, requiring a detailed model and a direct measurement of the disturbances (or a very accurate estimate) to be viable. This is unrealistic in most cases. (3) The measurement or estimation of V may be difficult to achieve (measuring vapor flow); thus, it may be sensitive to measurement error when trying to implement the optimal V. Thus, the approach here is to use self-optimizing control. As candidate measurements (y), we include all flows (ratioed to the feed), as well as the temperature at all stages. This gives ∼60 measurements. The component compositions in each product stream are also measured; however, because of the fact that they are active constraints (and, thus, are constant), they are not useful for self-optimizing control and are not included in y. Alternative 1: Two Degrees of Freedom. We first consider using the two available unconstrained degrees of freedom (nu ) 2) to control (and fix) two measurement combinations. The two unconstrained degrees of freedom could, for example, be u ) [RL RV] (but the specific choice does not actually matter for the steady-state analysis). The assumed disturbance vector is (these are determined to be the most important disturbances, from those listed in Table 1)
[]
zA z dtdf ) B q xB,S
(19)
where the subscript “tdf” denotes that the are two degrees of freedom. The last entry represents the composition offset for the side-stream product. The feed rate F is not included, because we have chosen to use only intensive variables when forming
852
Ind. Eng. Chem. Res., Vol. 46, No. 3, 2007
Table 3. Percentage Loss (in V) for All Disturbancesa Loss [%] Alternative
F-
F+
zA-
zA+
zB-
zB+
q-
q+
1 2 3 4
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0171 0.0037 5.0840 46.7037
0.0207 0.1340 11.8810 6.3019
0.0166 0.2247 0.3469 95.1660
0.0111 0.1666 0.8295 9.8256
0.0001 0.1876 1.0441 32.4629
0.0000 0.1084 1.1740 6.0578
Alternative
0 xA,D +
0 xA,D -
0 xC,B +
0 xC,B -
0 xB,S +
0 xB,S -
b (Lmax n )
c (Lavg n )
1 2 3 4
0.0025 0.0040 0.0074 0.0262
0.0095 0.0110 0.0207 0.0253
0.0639 0.0060 0.0033 0.0245
0.2082 0.0174 0.0034 0.0311
Loss [%]
b
0.0002 0.0004 0.0025 0.2579
0.0007 0.0004 0.0075 1.0198
0.0213 0.0847 0.2108 9.3142
0.0117 0.0206 0.0475 3.6254
a The symbol “-’’ in the subscript denotes negative perturbation, whereas “+’’ in the subscript denotes positive perturbation from the nominal value. Maximum loss for the implementation errors. c Average loss for the implementation errors.
the controlled variables (with a constant column efficiency, a feed rate change is automatically compensated for at steady state by fixing intensive variables). To use the null space method, we need from Theorem 1 to combine nu + nd ) 2 + 4 ) 6 measurements. To select the best subset of 6 out of the ∼60 candidate measurements, we use the measurement selection approach mentioned in the discussion section. This results in the following six temperature measurements:
[]
T37 T11 T ytdf ) T43 25 T4 T9
(20)
ctdf,1 ) -0.472T37 + 0.312T11 + 0.113T43 - 0.457T25 + 0.561T4 - 0.378T9 (21) ctdf,2 ) 0.185T37 + 0.376T11 - 0.667T43 - 0.524T25 0.154T4 + 0.285T9 (22) Alternative 2: One Degree of Freedom. Until now, we have assumed that the vapor split RV is a degree of freedom during operation (available for manipulation); however, most likely, this is not possible in practice. Therefore, it is interesting to consider the case where RV is fixed. In fixing RV , we add the implementation error of controlling RV to the disturbance vector and get
[]
[]
T37 T10 T yodf ) T43 27 T5 T12
(24)
which is very similar to the two-degrees-of-freedom case. The optimal measurement combination from the null space method is
codf ) -0.388T37 -0.658T10 + 0.192T43 - 0.0471T27 + 0.448T5 + 0.421T12 (25)
The location of the selected measurements is shown in Figure 3. Note that the majority of measurements are located in the bottom portion of the column, whereas only two measurements are located above the feed point. The sensitivity matrix F was obtained numerically by perturbing each of the four disturbances and reoptimizing. The null space method in Theorem 1 gives the optimal matrix H, corresponding to the following measurement combinations (controlled variables):
zA zB dodf ) q xB,S RV
The corresponding minimum number of measurements needed for the null space method is nu + nd ) 1 + 5 ) 6. The following subset of measurements was obtained:
(23)
Loss Evaluation Using a Nonlinear Model. The two previously mentioned control structures are compared with two alternative control structures (see Table 2). Alternative 3 is a control structure proposed by Halvorsen and Skogestad,20 where cDTS ) (T4 - T28) + (T12 - T36) is a symmetry measure of the temperature difference over the dividing wall, whereas alternative 4 is the “open loop” approach. The nonlinear losses for the alternative control structures for different realistic magnitude of the disturbances and measurement errors are given in Table 3. In the table, we have also included losses for changes (“disturbances’’) in the distillate and bottom product compositions (active constraints), which were not included in the original disturbances used to derive ctdf and codf.The conclusion is that the self-optimizing properties are excellent for both alternatives 1 and 2. When fixing two measurement combinations in alternative 1, the loss in energy usage (V) is
